A proof of the butterfly theorem using the scale factor between the two wings.

Martin Celli
October 21st 2016

Theorem (The Butterfly Theorem)

Let $M\;$ be the midpoint of a chord $PQ\;$ of a circle, through which two other chords $AB\;$ and $CD\;$ are drawn. Let us assume that $A\;$ and $D\;$ do not belong to a same half-plane defined by $PQ.\;$ Let $X\;$ (respectively $Y)\;$ be the intersection of $AD\;$ (respectively $BC)\;$ and $PQ.$

Butterfly theorem

Then $M\;$ is also the midpoint of $XY.$

Let $O\;$ be the center of the circle. The points $A\;$ and $C\;$ belong to a same half-plane defined by $PQ,\;$ the points $B\;$ and $D\;$ to the other half-plane. For sake of simplicity, we can assume that $O\;$ belongs to the same half-plane as $B\;$ and $D.$

In the proof, we directly show that the ratio $\displaystyle\frac{\sin(CYM)}{\sin(AXM)}\;$ is nothing but the scale factor between the two wings $AMD\;$ and $CMB,\;$ which are similar by the Inscribed Angle Theorem:

$\displaystyle\frac{\sin(CYM)}{\sin(AXM)}=\alpha,\;$ where $\displaystyle\alpha=\frac{CM}{AM}=\frac{BM}{DM}=\frac{CB}{AD}.$

More precisely, we have:

$\displaystyle\sin(AXM)=\frac{DM^2-AM^2}{AD\cdot OM}.$

As a matter of fact:

$\displaystyle\begin{align} DM^2-AM^2 &= ||\overrightarrow{DO}+\overrightarrow{OM}||^2-||\overrightarrow{AO}+\overrightarrow{OM}||^2\\ &=OD^2+OM^2+2\overrightarrow{DO}\cdot\overrightarrow{OM}-(OA^2+OM^2+2\overrightarrow{AO}\cdot\overrightarrow{OM})\\ &=2\overrightarrow{DO}\cdot\overrightarrow{OM}\; \text{since }OA=OD\\ &=2AD\cdot OM\sin(AXM)\;\text{as } OMX=OMY=\frac{MX+OMY}{2}=\frac{\pi}{2}, \end{align}$

because triangle $OMP\;$ and $OMQ\;$ are congruent. Similarly, we have:

$\displaystyle\sin(CYM)=\frac{BM^2-CM^2}{CB\cdot OM}.$

Thus:

$\displaystyle\begin{align} \frac{XM}{YM}&=\frac{AM}{CM}\cdot\frac{CM}{YM}\cdot\frac{XM}{AM}\\ &=\frac{AM}{CM}\cdot\frac{\sin(CYM)}{\sin(YCM)}\cdot\frac{\sin(XAM)}{\sin(AXM)} \end{align}$

by the law of sines, applied in triangles $AXM\;$ and $CYM.\;$ Now we continue using the Inscribed Angle Theorem, as the next step:

$\displaystyle\begin{align} \frac{XM}{YM}&=\frac{AM}{CM}\cdot\frac{CM}{YM}\cdot\frac{XM}{AM}\\ &=\frac{AM}{CM}\cdot\frac{\sin(CYM)}{\sin(AXM)}\\ &=\frac{AM}{CM}\cdot\frac{BM^-CM^2}{CB\cdot OM}\cdot\frac{AD\cdot OM}{DM^2-AM^2}\\ &=\frac{AM}{CM}\cdot\frac{AD}{CB}\cdot\frac{BM^2-CM^2}{DM^2-AM2}\\ &=\frac{1}{\alpha}\cdot\frac{1}{\alpha}\cdot\alpha^2\\ &=1. \end{align}$

Butterfly Theorem and Variants

  1. Butterfly theorem
  2. 2N-Wing Butterfly Theorem
  3. Better Butterfly Theorem
  4. Butterflies in Ellipse
  5. Butterflies in Hyperbola
  6. Butterflies in Quadrilaterals and Elsewhere
  7. Pinning Butterfly on Radical Axes
  8. Shearing Butterflies in Quadrilaterals
  9. The Plain Butterfly Theorem
  10. Two Butterflies Theorem
  11. Two Butterflies Theorem II
  12. Two Butterflies Theorem III
  13. Algebraic proof of the theorem of butterflies in quadrilaterals
  14. William Wallace's Proof of the Butterfly Theorem
  15. Butterfly theorem, a Projective Proof
  16. Areal Butterflies
  17. Butterflies in Similar Co-axial Conics
  18. Butterfly Trigonometry
  19. Butterfly in Kite
  20. Butterfly with Menelaus
  21. William Wallace's 1803 Statement of the Butterfly Theorem
  22. Butterfly in Inscriptible Quadrilateral
  23. Camouflaged Butterfly
  24. General Butterfly in Pictures
  25. Butterfly via Ceva
  26. Butterfly via the Scale Factor of the Wings
  27. Butterfly by Midline
  28. Stathis Koutras' Butterfly
  29. The Lepidoptera of the Circles
  30. The Lepidoptera of the Quadrilateral
  31. The Lepidoptera of the Quadrilateral II
  32. The Lepidoptera of the Triangle
  33. Two Butterflies Theorem as a Porism of Cyclic Quadrilaterals
  34. Two Butterfly Theorems by Sidney Kung

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